Tunable topological phononic crystals

Zeguo Chen (陈泽国) and Ying Wu (吴莹) King Abdullah University of Science and Technology

12  Jan  2016,  Hong  Kong  

Outline

•  Introduction – Background and motivation

•  Tunable topological phononic crystals – Design – Physical model – Topological properties – Demonstration

•  Summary

Introduction

Nat.  Phys.  11,  799  (2015)   Rev.  Mod.  Phys.    82  3045  (2010)  

One  way  propagaFon  edge  states.    

•  QH state

•  QSH states

TKNN invariant

•  Chern number. –  Arises from topology that differentiates QH (non-trivial)

and an ordinary insulator (trivial). –  Integration of Berry flux in the BZ.

Cm =12π

d 2∫k∇×

Am

Am = i um ∇k umBerry  connecFon  

Nat.  Photonics    8    821  (2014)  

Classical analogues in photonic systems

Phys.  Rev.  LeK.  100,  013905  (2008)  Phys.  Rev.  LeK.  100,  013904  (2008).      

YIG  material  

About acoustics

•  Breaking Time-reversal symmetry in acoustics

Nonreciprocal air-flow-contained acoustic circulator

Science  343,  516  (2014)    

About acoustics

PRL 114, 114301 (2015)  

NJP  17,  053016  (2015).  

Nat.  Commun.  6,  8260  (2015).  

Motivation

•  From deterministic degeneracy at K point to accidental degeneracy at point Γ

Protected  by  TRS  

Accidental  degeneracy  

What  happens  if    Fme-­‐reversal  symmetry  is  broken  and  ,at  the  same  Fme,  geometry  changes  as  well?  

PRB  86  035141  (2012)  

Sample

0r1r d

0 0.35r m=

1 0.5r m=

2a m=

−ρc2iω(iωφ + v ⋅∇φ)+∇⋅ (ρ∇φ − ρ

c2(iωφ + v ⋅∇φ)v) = 0

v = 0No  air  flow  

22 ( ) 0cρω φ ρ φ+∇⋅ ∇ =

100

120

140

160

180

200

ΜΧΓΜ

Frequency(Hz)

0 0.07336d m=

100

120

140

160

180

200

Frequency(Hz)

ΜΧΓΜ

Without air flow

x y 2 2x y−

gap  

0d0d d< 0d d>

0.04 0.06 0.08 0.10125

130

135

140

145

150

ϕpx, ϕpy ϕd

Freq

uenc

y(H

z)

d(m)

d1 d0 d2

Gap  11 12 12

12 22 22

12 22 22

2 (cos cos ) 2 sin 2 sin2 sin 2 cos 2 cos 02 sin 0 2 cos 2 cos

d x x y x x x y

x x px x x y y

x y py y x x y

E t k a k a it k a it k aH it k a E t k a t k a

it k a E t k a t k a

⎡ ⎤+ +⎢ ⎥= − + +⎢ ⎥⎢ ⎥− + +⎣ ⎦

tmij = Φi (

r ) H Φ j (r + rm )

A tight-binding model

k = 0

11

22 22

22 22

2 0 00 2 2 00 0 2 2

d x

px x y

py y x

E tH E t t

E t t

⎡ ⎤+⎢ ⎥= + +⎢ ⎥⎢ ⎥+ +⎣ ⎦

mainly  contributed  by  the  wave  funcFons  inside  the  waveguide  

E  is  the  on-­‐site  energy  of  the  rings  

Eigenvalue  depends  on  the  width  of  the  waveguide.  

Breaking T Symmetry: with air flow

0 5 10 15130

132

134

136

138

140

142

Freq

uenc

y(H

z)

v (m/s)

ϕd

ϕpx

ϕpyv(x, y) = ( −vy

x2 + y2, vx

x2 + y2) = veθ

( ) avc v Rω± = ± ( )0 1 2avR r r= +

From  C4v  to  C4.                      almost  does  not  change  ϕd

Degeneracy  is  li`ed.  

d = d0

Band structures

Χ ΜΓΜΧΓ

100

120

140

160

180

200

Frequency(Hz)

ΧΓ ΜΜ

•  Degeneracy  associated  with              and              li`s.    

•  Branch  associated  with            almost  does  not  change.  

Consistent  with  the  Tight-­‐Binding.  

pxϕ pyϕ

dϕ

0.04 0.06 0.08 0.10125

130

135

140

145

150

Freq

uenc

y(H

z)

d(m)

ϕpx

ϕpy

ϕd

d1dt d2

0.04 0.06 0.08 0.10125

130

135

140

145

150

ϕpx, ϕpy ϕd

Freq

uenc

y(H

z)

d(m)

d1 d0 d2

Topological property: Chern number

C = i2π

∇ k × un (k )

BZ∫n∑ ∇ k un (

k ) d 2

k

( ) ( ) (Y)Cj j j

j

i ξ ξ ζ= Γ Μ∏

Phys.  Rev.  B  86,  115112  (2012)    

0.04 0.06 0.08 0.10125

130

135

140

145

150

Freq

uenc

y(H

z)

d(m)

ϕpx

ϕpy

ϕd

d1dt d2

100

120

140

160

180

200

Frequency(Hz)

0

ΧΓ ΜΜ

0

Trivial  

100

120

140

160

180

200

ΜΧΓΜ

Frequency(Hz)

1

0

Non-­‐trivial  

100

120

140

160

180

200

2

-1

ΧΜ Μ

Frequency(Hz)

Γ

Trivial  Non-­‐trivial  

TransiFon  point  

TransiFon  point  depends  on:    geometry  &  intensity  of  the  flow  

Tunable!    

Topological properties

C=0   C=1   C=1  

d  =  0.1m  

Edge states

 B                          A  

Demonstration I: fixed air flow

One-­‐way  propagaFon  (defect  immune)  

Demonstration I: fixed air flow

td d<

td d>

Interface  state  

Demonstration II: fixed geometry

Trivial  

Non-­‐trivial  

0 5 10 15137

138

139

Freq

uenc

y(H

z)

v (m/s)

ϕd

ϕpy

0.065d m=

Demonstration II: fixed geometry

5 /v m s=

15 /v m s=

Summary

•  Designed a topological phononic crystal •  Topological property depends on both the

geometry and time-reversal symmetry •  One-way propagation edge state is

observed. •  Tunable property is demonstrated.

Acknowledgement:  •  KAUST  baseline  research  fund.      

Thank you. More  informaFon:      hKp://arxiv.org/abs/1512.00814  ying.wu@kaust.edu.sa  chen.zeguo@kaust.edu.sa